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A261400
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Number of n X n knot mosaics.
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8
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1, 2, 22, 2594, 4183954, 101393411126, 38572794946976686, 234855052870954505606714, 23054099362200397056093750003442, 36564627559441095000442883434988307728126, 937273142571326346553334567317274833729462713413038
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OFFSET
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1,2
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COMMENTS
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According to Oh, Hong, Lee, and Lee, a(n) grows at a quadratic exponential rate. Moreover, it appears that the ratios A374947(n)/a(n) converge to 0 at a quadratic exponential rate. - Luc Ta, Aug 27 2024
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LINKS
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Hwa Jeong Lee, Kyungpyo Hong, Ho Lee, and Seungsang Oh, Mosaic number of knots, arXiv: 1301.6041 [math.GT], 2014.
Samuel J. Lomonaco and Louis H. Kauffman, Quantum Knots and Mosaics, Proc. Sympos. Applied Math., Amer. Math. Soc., Vol. 68 (2010), pp. 177-208.
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MATHEMATICA
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x[0] = o[0] = {{1}};
x[n_] := ArrayFlatten[{{x[n - 1], o[n - 1]}, {o[n - 1], x[n - 1]}}];
o[n_] := ArrayFlatten[{{o[n - 1], x[n - 1]}, {x[n - 1], 4*o[n - 1]}}];
mosaicsSquare[n_] := If[n > 1, 2*Total[MatrixPower[x[n - 2] + o[n - 2], n - 2], 2], 1];
Flatten[ParallelTable[mosaicsSquare[n], {n, 1, 11}]] (* This program is based on Theorem 1 of Oh, Hong, Lee, and Lee (see Links). - Luc Ta, Aug 13 2024 *)
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CROSSREFS
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Reminiscent of (but of course different from) A200000.
The term 22 is the same 22 that appears in A261399.
a(n) is the main diagonal of A375353.
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KEYWORD
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nonn,changed
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AUTHOR
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EXTENSIONS
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Typo in a(11) corrected by Luc Ta, Aug 13 2024
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STATUS
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approved
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